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Signal models are a cornerstone of contemporary signal and image processing methodology. In this report, two particular signal modeling methods, called analysis and synthesis sparse representation, are studied which have been proven to be effective for many signals, such as natural images, and successfully used in a wide range of applications. Both models represent signals in terms of linear combinations of an underlying set, called dictionary, of elementary signals known as atoms. The driving force behind both models is sparsity of the representation coefficients, i.e. the rapid decay of the representation coefficients over the dictionary. On the other hands, the dictionary choice determines the success of the entire model. According to these two signal models, there have been two main disciplines of dictionary designing; harmonic analysis approach and machine learning methodology. The former leads to designing the dictionaries with easy and fast implementation, while the latter provides a simple and expressive structure for designing adaptable and efficient dictionaries. The main goal of this thesis is to provide new applications to these signal modeling methods by addressing several problems from various perspectives. It begins with the direct application of the sparse representation, i.e. image compression. The line of research followed in this area is the synthesis-based sparse representation approach in the sense that the dictionary is not fixed and predefined, but learned from training data and adapted to data, yielding a more compact representation. A new Image codec based on adaptive sparse representation over a trained dictionary is proposed, wherein different sparsity levels are assigned to the image patches belonging to the salient regions, being more conspicuous to the human visual system. Experimental results show that the proposed method outperforms the existing image coding standards, such as JPEG and JPEG2000, which use an analytic dictionary, as well as the state-of-the-art codecs based on the trained dictionaries. In the next part of thesis, it focuses on another important application of the sparse signal modeling, i.e. solving inverse problems, especially for error concealment (EC), wherein a corrupted image is reconstructed from the incomplete data, and Compressed Sensing recover, where an image is reconstructed from a limited number of random measurements. Signal modeling is usually used as a prior knowledge about the signal to solve these NP-hard problems. In this thesis, inspired by the analysis and synthesis sparse models, these challenges are transferred into two distinct sparse recovery frameworks and several recovery methods are proposed. Compared with the state-of-the-art EC and CS algorithms, experimental results show that the proposed methods show better reconstruction performance in terms of objective and subjective evaluations. This thesis is finalized by giving some conclusions and introducing some lines for future works.; La modélisation des signaux peut être vue comme la pierre angulaire de la méthodologie contemporaine de traitement du signal et de l'image. La modélisation parcimonieuse permets la représentation des signaux en termes de combinaisons linéaires d'un ensemble sous-jacent, appelé dictionnaire, de signaux élémentaires connus sous le nom d'atomes. La force motrice de ce modèle est la rareté des coefficients de représentation, c'est-à-dire la décroissance rapide des coefficients de représentation sur le dictionnaire. L'objectif principal de cette thèse est de fournir de nouvelles applications pour cette méthode de modélisation du signal en abordant plusieurs problèmes sous différents angles. On se concentre sur une autre application importante de la modélisation parcimonieuse des signaux, à savoir la résolution des problèmes inverses, notamment la compensation des erreurs, la reconstruction des images incomplètes et la reconstruction des images compresses à partir d'un nombre limité de mesures aléatoires. La modélisation du signal est généralement utilisée comme une connaissance préalable du signal pour résoudre ces problèmes NP-difficiles. Puis, Il commence par l'application directe de la représentation éparse, c'est-à-dire a la compression d'image. Un nouveau codec image basé sur la représentation éparse adaptative sur un dictionnaire formé est proposé, dans lesquels différents niveaux de densité sont assignés aux correctifs d'image appartenant aux régions saillantes. Dans cette thèse, ces défis sont transférés dans des cadres distincts d’acquisition comprimée et plusieurs méthodes de reconstruction sont proposées.

Signal models are a cornerstone of contemporary signal and image processing methodology. In this report, two particular signal modeling methods, called analysis and synthesis sparse representation, are studied which have been proven to be effective for many signals, such as natural images, and successfully used in a wide range of applications. Both models represent signals in terms of linear combinations of an underlying set, called dictionary, of elementary signals known as atoms. The driving force behind both models is sparsity of the representation coefficients, i.e. the rapid decay of the representation coefficients over the dictionary. On the other hands, the dictionary choice determines the success of the entire model. According to these two signal models, there have been two main disciplines of dictionary designing; harmonic analysis approach and machine learning methodology. The former leads to designing the dictionaries with easy and fast implementation, while the latter provides a simple and expressive structure for designing adaptable and efficient dictionaries. The main goal of this thesis is to provide new applications to these signal modeling methods by addressing several problems from various perspectives. It begins with the direct application of the sparse representation, i.e. image compression. The line of research followed in this area is the synthesis-based sparse representation approach in the sense that the dictionary is not fixed and predefined, but learned from training data and adapted to data, yielding a more compact representation. A new Image codec based on adaptive sparse representation over a trained dictionary is proposed, wherein different sparsity levels are assigned to the image patches belonging to the salient regions, being more conspicuous to the human visual system. Experimental results show that the proposed method outperforms the existing image coding standards, such as JPEG and JPEG2000, which use an analytic dictionary, as well as the state-of-the-art codecs based on the trained dictionaries. In the next part of thesis, it focuses on another important application of the sparse signal modeling, i.e. solving inverse problems, especially for error concealment (EC), wherein a corrupted image is reconstructed from the incomplete data, and Compressed Sensing recover, where an image is reconstructed from a limited number of random measurements. Signal modeling is usually used as a prior knowledge about the signal to solve these NP-hard problems. In this thesis, inspired by the analysis and synthesis sparse models, these challenges are transferred into two distinct sparse recovery frameworks and several recovery methods are proposed. Compared with the state-of-the-art EC and CS algorithms, experimental results show that the proposed methods show better reconstruction performance in terms of objective and subjective evaluations. This thesis is finalized by giving some conclusions and introducing some lines for future works., La modélisation des signaux peut être vue comme la pierre angulaire de la méthodologie contemporaine de traitement du signal et de l'image. La modélisation parcimonieuse permets la représentation des signaux en termes de combinaisons linéaires d'un ensemble sous-jacent, appelé dictionnaire, de signaux élémentaires connus sous le nom d'atomes. La force motrice de ce modèle est la rareté des coefficients de représentation, c'est-à-dire la décroissance rapide des coefficients de représentation sur le dictionnaire. L'objectif principal de cette thèse est de fournir de nouvelles applications pour cette méthode de modélisation du signal en abordant plusieurs problèmes sous différents angles. On se concentre sur une autre application importante de la modélisation parcimonieuse des signaux, à savoir la résolution des problèmes inverses, notamment la compensation des erreurs, la reconstruction des images incomplètes et la reconstruction des images compresses à partir d'un nombre limité de mesures aléatoires. La modélisation du signal est généralement utilisée comme une connaissance préalable du signal pour résoudre ces problèmes NP-difficiles. Puis, Il commence par l'application directe de la représentation éparse, c'est-à-dire a la compression d'image. Un nouveau codec image basé sur la représentation éparse adaptative sur un dictionnaire formé est proposé, dans lesquels différents niveaux de densité sont assignés aux correctifs d'image appartenant aux régions saillantes. Dans cette thèse, ces défis sont transférés dans des cadres distincts d’acquisition comprimée et plusieurs méthodes de reconstruction sont proposées.

In passive monitoring using sensor networks, low energy supplies drastically constrain sensors in terms of calculation and communication abilities. Designing processing algorithms at the sensor level that take into account these constraints is an important problem in this context. We study here the estimation of correlation functions between sensors using compressed acquisition and one-bit-quantization. The estimation is achieved directly using compressed samples, without considering any reconstruction of the signals. We show that if the signals of interest are far from white noise, estimation of the correlation using $M$ compressed samples out of $N\geq M$ can be more advantageous than estimation of the correlation using $M$ consecutive samples. The analysis consists of studying the asymptotic performance of the estimators at a fixed compression rate. We provide the analysis when the compression is realized by a random projection matrix composed of independent and identically distributed entries. The framework includes widely used random projection matrices, such as Gaussian and Bernoulli matrices, and it also includes very sparse matrices. However, it does not include subsampling without replacement, for which a separate analysis is provided. When considering one-bit-quantization as well, the theoretical analysis is not tractable. However, empirical evidence allows the conclusion that in practical situations, compressed and quantized estimators behave sufficiently correctly to be useful in, for example, time-delay estimation and model estimation., submitted

This thesis is dedicated to the application of the novel compressed sensing theory to the acquisition and reconstruction of 3D US images and Doppler signals. In 3D US imaging, one of the major difficulties concerns the number of RF lines that has to be acquired to cover the complete volume. The acquisition of each line takes an incompressible time due to the finite velocity of the ultrasound wave. One possible solution for increasing the frame rate consists in reducing the acquisition time by skipping some RF lines. The reconstruction of the missing information in post processing is then a typical application of compressed sensing. Another excellent candidate for this theory is the Doppler duplex imaging that implies alternating two modes of emission, one for B-mode imaging and the other for flow estimation. Regarding 3D imaging, we propose a compressed sensing framework using learned overcomplete dictionaries. Such dictionaries allow for much sparser representations of the signals since they are optimized for a particular class of images such as US images.We also focus on the measurement sensing setup and propose a line-wise sampling of entire RF lines which allows to decrease the amount of data and is feasible in a relatively simple setting of the 3D US equipment. The algorithm was validated on 3D simulated and experimental data. For the Doppler application, we proposed a CS based framework for randomly interleaving Doppler and US emissions. The proposed method reconstructs the Doppler signal using a block sparse Bayesian learning algorithm that exploits the correlation structure within a signal and has the ability of recovering partially sparse signals as long as they are correlated. This method is validated on simulated and experimental Doppler data.; DL’objectif de cette thèse est le développement de techniques adaptées à l’application de la théorie de l’acquisition compressée en imagerie ultrasonore 3D et Doppler. En imagerie ultrasonore 3D une des principales difficultés concerne le temps d’acquisition très long lié au nombre de lignes RF à acquérir pour couvrir l’ensemble du volume. Afin d’augmenter la cadence d’imagerie une solution possible consiste à choisir aléatoirement des lignes RF qui ne seront pas acquises. La reconstruction des données manquantes est une application typique de l’acquisition compressée. Une autre application d’intérêt correspond aux acquisitions Doppler duplex où des stratégies d’entrelacement des acquisitions sont nécessaires et conduisent donc à une réduction de la quantité de données disponibles. Dans ce contexte, nous avons réalisé de nouveaux développements permettant l’application de l’acquisition compressée à ces deux modalités d’acquisition ultrasonore. Dans un premier temps, nous avons proposé d’utiliser des dictionnaires redondants construits à partir des signaux d’intérêt pour la reconstruction d’images 3D ultrasonores. Une attention particulière a aussi été apportée à la configuration du système d’acquisition et nous avons choisi de nous concentrer sur un échantillonnage des lignes RF entières, réalisable en pratique de façon relativement simple. Cette méthode est validée sur données 3D simulées et expérimentales. Dans un deuxième temps, nous proposons une méthode qui permet d’alterner de manière aléatoire les émissions Doppler et les émissions destinées à l’imagerie mode-B. La technique est basée sur une approche bayésienne qui exploite la corrélation et la parcimonie des blocs du signal. L’algorithme est validé sur des données Doppler simulées et expérimentales.

This thesis is dedicated to the application of the novel compressed sensing theory to the acquisition and reconstruction of 3D US images and Doppler signals. In 3D US imaging, one of the major difficulties concerns the number of RF lines that has to be acquired to cover the complete volume. The acquisition of each line takes an incompressible time due to the finite velocity of the ultrasound wave. One possible solution for increasing the frame rate consists in reducing the acquisition time by skipping some RF lines. The reconstruction of the missing information in post processing is then a typical application of compressed sensing. Another excellent candidate for this theory is the Doppler duplex imaging that implies alternating two modes of emission, one for B-mode imaging and the other for flow estimation. Regarding 3D imaging, we propose a compressed sensing framework using learned overcomplete dictionaries. Such dictionaries allow for much sparser representations of the signals since they are optimized for a particular class of images such as US images.We also focus on the measurement sensing setup and propose a line-wise sampling of entire RF lines which allows to decrease the amount of data and is feasible in a relatively simple setting of the 3D US equipment. The algorithm was validated on 3D simulated and experimental data. For the Doppler application, we proposed a CS based framework for randomly interleaving Doppler and US emissions. The proposed method reconstructs the Doppler signal using a block sparse Bayesian learning algorithm that exploits the correlation structure within a signal and has the ability of recovering partially sparse signals as long as they are correlated. This method is validated on simulated and experimental Doppler data., DL’objectif de cette thèse est le développement de techniques adaptées à l’application de la théorie de l’acquisition compressée en imagerie ultrasonore 3D et Doppler. En imagerie ultrasonore 3D une des principales difficultés concerne le temps d’acquisition très long lié au nombre de lignes RF à acquérir pour couvrir l’ensemble du volume. Afin d’augmenter la cadence d’imagerie une solution possible consiste à choisir aléatoirement des lignes RF qui ne seront pas acquises. La reconstruction des données manquantes est une application typique de l’acquisition compressée. Une autre application d’intérêt correspond aux acquisitions Doppler duplex où des stratégies d’entrelacement des acquisitions sont nécessaires et conduisent donc à une réduction de la quantité de données disponibles. Dans ce contexte, nous avons réalisé de nouveaux développements permettant l’application de l’acquisition compressée à ces deux modalités d’acquisition ultrasonore. Dans un premier temps, nous avons proposé d’utiliser des dictionnaires redondants construits à partir des signaux d’intérêt pour la reconstruction d’images 3D ultrasonores. Une attention particulière a aussi été apportée à la configuration du système d’acquisition et nous avons choisi de nous concentrer sur un échantillonnage des lignes RF entières, réalisable en pratique de façon relativement simple. Cette méthode est validée sur données 3D simulées et expérimentales. Dans un deuxième temps, nous proposons une méthode qui permet d’alterner de manière aléatoire les émissions Doppler et les émissions destinées à l’imagerie mode-B. La technique est basée sur une approche bayésienne qui exploite la corrélation et la parcimonie des blocs du signal. L’algorithme est validé sur des données Doppler simulées et expérimentales.